Question

A uranium nucleus (238U) moves along the positive 𝑥-axis with speed 𝑣 = 5.0 × 105 m/s when it decays
into an 𝛼-particle (4He, helium nucleus) and a thorium nucleus (234Th). If 4He moves off at an angle of 20.2° above the 𝑥-axis with a speed 𝑣𝐻𝑒 = 1.4 × 10^7 m/s, you will determine what is the recoil velocity, 𝑣 , of 234Th and the angle below the 𝑥-axis. Assume the uranium-thorium-alpha system is isolated with not external forces acting on it and the masses of are A × 1.673 × 10^−27 kg. This is a two-dimensional problem.

Answer 1

You did not type clearly what the masses are
call mass U238 = mu
call mass Th = mt
call mass alpha = ma
speed of Th = st
speed of U = su
speed of alpha = 1.4*10^7
hopefully the Helium mass + the Thorium mass is the Uranium mass
then momentum is conserved
A is angle below x axis of Th
x direction
mu * 5*10^5 = ma * 1.4*10^7 * cos 20.2 + mt * st * cos A
y direction
0 = ma * 1.4*10^7 * sin 20.2 + mt* st * sin A
solve for the Th speed st and Angle A below x axis